where \(\rho \)  is the density and \(\mathbf {u}=(u_x,u_y)\)  is the velocity of the fluid, and \(c_s\)  is the speed of sound, which is a free parameter and will be related to the transport coefficients via a C-E analysis later. It may be noted that unlike the prior rectangular MRT-LB schemes that use equilibrium moments with fluid velocity terms only up to the second order, the above moment equilibria (Eq. (REF )) involve the use of higher order velocity terms arising naturally via the use of the matching principle noted above, which preserves the GI of the moments independently supported by the lattice aside from those subjected to the aliasing effects. In addition to avoiding the orthogonalization in defining the moment basis, consideration of such more refined equilibria is expected to yield a more robust rectangular LB formulation. The components of the source moments that would yield consistency with the NS equations are given by [1]}
\(\Phi _0 &=& \sigma _{0}=0,\nonumber \\\Phi _1 &=& \sigma _{x}=F_x,\nonumber \\\Phi _2 &=& \sigma _{y}=F_y,\nonumber \\\Phi _3 &=& \sigma _{xx+yy}=2(F_x u_x+F_y u_y),\nonumber \\\Phi _4 &=& \sigma _{xx-yy}=2(F_x u_x-F_y u_y),\nonumber \\\Phi _5 &=& \sigma _{xy}=F_xu_y+F_yu_x,\nonumber \\\Phi _6 &=& \sigma _{xxy}=F_yu_x^2+2F_xu_xu_y,\nonumber \\\Phi _7 &=& \sigma _{xyy}=F_xu_y^2+2F_yu_yu_x,\nonumber \\\Phi _8 &=& \sigma _{xxyy}=2(F_xu_xu_y^2+F_yu_yu_x^2),\) 
